Methods and apparatus to point a payload at a target

ABSTRACT

An example machine accessible medium having instructions stored thereon that, when executed, causes a machine to at least command a first actuator to move to a first corrected stroke position and a second actuator to move to a second corrected stroke position to point a payload at a target along a line of sight vector without verifying a target pointing direction of a base when the first and second actuators are positioned to the respective first and second corrected stroke positions and without using a feedback to verify the base being at the target pointing direction when the first and second actuators are positioned to the respective first and second corrected stroke positions.

CROSS-REFERENCE TO RELATED APPLICATIONS

The patent arises from a continuation of U.S. patent application Ser.No. 13/665,857, filed on Oct. 31, 2012, titled Methods and Apparatus toPoint a Payload at a Target, and is hereby incorporated herein byreference in its entirety.

FIELD

The present disclosure relates generally to payloads and, moreparticularly, to methods and apparatus to point a payload at a target.

BACKGROUND

Generally, a payload is to be substantially pointed at a target. Forexample, an antenna is to be substantially pointed at a target to enablecommunication between the antenna and the target. If the antenna pointsaway from the target, the communication between the antenna and thetarget is affected. The antenna may be disposed on a satellite in orbitaround Earth. Due to the distance between the satellite and a target onEarth, an alignment error of the antenna influenced by thermaldistortion, machining tolerances, etc. may cause the antenna to pointaway from the target.

SUMMARY

An example machine accessible medium having instructions stored thereonthat, when executed, causes a machine to at least: estimate a targetpointing direction of a base supporting a payload to point the payloadat a target, where the base is movable relative to a pivot about anazimuth angle and an elevation angle via a first actuator and a secondactuator directly coupled to the base; determine a first estimatedstroke position of the first actuator and a second estimated strokeposition of a second actuator to position the base at the targetpointing direction. The instructions further cause the machine to obtaina base orientation error. The instructions further cause the machine todetermine a first stroke position error of the first actuator and asecond stroke position error of the second actuator based on the baseorientation error. The instructions further cause the machine todetermine a first corrected stroke position of the first actuator basedon a difference between the first stroke position error and the firstestimated stroke position. The instructions further cause the machine todetermine a second corrected stroke position of the second actuatorbased on a difference between the second stroke position error and thesecond estimated stroke position. Also, the instructions cause themachine to and command the first actuator to move to the first correctedstroke position and the second actuator to move to the second correctedstroke position to point the payload at the target along a line of sightvector without verifying the target pointing direction of the base whenthe first and second actuators are positioned to the respective firstand second corrected stroke positions and without using a feedback toverify the base being at the target pointing direction when the firstand second actuators are positioned to the respective first and secondcorrected stroke positions.

Another example tangible machine readable storage medium disclosedherein includes instructions that, when executed, cause a machine to atleast receive a first command to move a payload relative to a firsttarget. The instructions also cause the machine to estimate a baseorientation of a base coupled to the payload to point the payload to thefirst target along a first line of sight vector. The instructions alsocause the machine to determine a first estimated stroke position of afirst linear actuator and a second estimated stroke position of a secondlinear actuator to position the base to the estimated base orientation.The instructions further cause the machine to modify the first estimatedstroke position to a first corrected stroke position based on a baseorientation error. Also, the instructions cause the machine to modifythe second estimated stroke position to a second corrected strokeposition based on the base orientation error. The instructions furthercause the machine to command the first linear actuator to actuate to thefirst corrected stroke position and commanding the second linearactuator to actuate to the second corrected stroke position withoutverifying a final position of the payload after the first linearactuator is actuated to the first corrected stroke position and thesecond linear actuator is actuated to the second corrected strokeposition.

Another example tangible machine readable storage medium disclosedherein includes instructions that, when executed, cause a machine to atleast receive a command to actuate a first linear actuator and a secondlinear actuator operatively coupled to a base of a payload to orient thebase at a first azimuth angle and a first elevation angle to point thepayload at a target, the base being pivotably coupled to the payload viaa joint and having only the first and second linear actuators to movethe base about the joint relative to the azimuth angle and the elevationangle; determine a pointing error based on an experimentally determinedsecond azimuth angle and an experimentally determined second elevationangle of the base to point the payload at the target; determine a firststroke position error of the first linear actuator and a second strokeposition error of the second linear actuator based on the pointingerror; determine a first corrected stroke position of the first linearactuator and a second corrected stroke position of the second linearactuator; and command the first linear actuator to move to the firstcorrected stroke position and the second linear actuator to move to thesecond corrected stroke position to point the payload at the targetwithout verifying a final position of the payload after the first linearactuator is actuated to the first corrected stroke position and thesecond linear actuator is actuated to the second corrected strokeposition

An example apparatus disclosed herein includes an instruction processorto receive a command to point a payload coupled to a base at a target,the base being movable relative to a pivot about an azimuth angle and anelevation angle via only a first linear actuator and a second linearactuator, the pivot, the first linear actuator and the second linearactuator being directly coupled to the base. In some examples, theapparatus includes a line of sight determiner to determine a line ofsight vector between the payload and the target. In some examples, theapparatus includes a base orientation determiner to determine anestimated base orientation to point the payload at the target. In someexamples, the apparatus includes an estimated stroke position determinerto determine a first estimated stroke position of a first linearactuator and a second estimated stroke position of a second linearactuator to orient the base at the estimated base orientation. In someexamples, the apparatus includes a base orientation error determiner todetermine a base orientation error. In some examples, the apparatusincludes a stroke position error determiner to determine a first strokepositioner error of the first linear actuator based on the baseorientation error and a second stroke position error of the secondlinear actuator based on the base orientation error. In some examplesthe apparatus includes a corrected stroke position determiner todetermine a first corrected stroke position of the first actuator and asecond corrected stroke position of the second actuator. In someexamples, the apparatus includes an actuator controller to command thefirst linear actuator to move to the first corrected stroke position andthe second linear actuator to move to the second corrected strokeposition to point the payload at the target along a line of sight vectorwithout verifying the target pointing direction of the base when thefirst and second linear actuators are positioned to the respective firstand second corrected stroke positions and without using a feedback toverify the base being at the target pointing direction when the firstand second linear actuators are positioned to the respective first andsecond corrected stroke positions.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example satellite in communication with a targetin accordance with the teachings of this disclosure.

FIG. 2 is a perspective view of an example payload assembly disclosedherein.

FIG. 3 is a rear view of the example payload assembly of FIG. 2.

FIG. 4 is a top view of the example payload assembly of FIGS. 2-3.

FIG. 5 is a two-dimensional plot of payload signal power levels detectedat the example target of FIG. 1.

FIG. 6 is a three-dimensional plot of the payload signal power levelsdetected at the example target of FIG. 1.

FIG. 7 is a block diagram of an example controller in accordance withthe teachings of this disclosure.

FIG. 8 is a flow diagram of an example process disclosed herein.

FIG. 9 is a flow diagram of another example process disclosed herein.

FIG. 10 is a block diagram of an example processing platform capable ofexecuting machine readable instructions to implement the examplecontroller of FIG. 7.

Wherever possible, the same reference numbers will be used throughoutthe drawing(s) and accompanying written description to refer to the sameor like parts. As used in this disclosure, stating that any part (e.g.,a layer, film, area, or plate) is in any way positioned on (e.g.,positioned on, located on, disposed on, or formed on, etc.) anotherpart, means that the referenced part is either in contact with the otherpart, or that the referenced part is above the other part with one ormore intermediate part(s) located therebetween. Stating that any part isin contact with another part means that there is no intermediate partbetween the two parts.

DESCRIPTION

Methods and apparatus to point a payload at a target are disclosedherein. An example apparatus disclosed herein includes a payload suchas, for example, an antenna, a transmitter, a sensor (e.g., an infraredsensor), an optical device, a camera, and/or any other suitable payloadcoupled to a base that is rotatable about a joint. In some examples, thebase is operatively coupled to a first actuator (e.g., a linear actuatorsuch as, for example, a jackscrew) and a second actuator (e.g., a linearactuator such as, for example, a jackscrew). The first actuator and thesecond actuator may enable rotation of the base about the joint toadjust an azimuth angle and/or an elevation angle of the base. Acontroller may be in communication with the first actuator and thesecond actuator to control a first stroke position of the first actuatorand a second stroke position of the second actuator.

To point the payload at a target, the controller may determine andcompensate for a first stroke position error corresponding to the firstactuator and a second stroke position error corresponding to the secondactuator. The first stroke position error and/or the second strokeposition error may be influenced by thermal distortion, machiningtolerances, alignment errors, etc. In some examples, the first strokeposition error and the second stroke position error are constantrelative to an orientation of the base and/or an amount of rotation ofthe base. As such, the base may be rotated over a wide distance range(e.g., ten degrees of rotation or more) via the first actuator and thesecond actuator to point the payload at the target.

FIG. 1 illustrates an example satellite 100 in accordance with theteachings of this disclosure. In the illustrated example, the satellite100 is in orbit around Earth 102. However, in other examples, thesatellite 100 may be in orbit around another celestial body such as, forexample, Earth's moon. In the illustrated example, the satellite 100 isin communication with a ground station 104 located on Earth 102. In someexamples, the satellite 100 is in communication with one or more otherground stations, satellites, and/or other targets. As described ingreater detail below, the example satellite 100 includes a payload (FIG.2), which may be employed to communicate information to the groundstation 104, receive signals, generate images, etc.

FIG. 2 is a perspective view of an example payload assembly 200disclosed herein, which may be used to communicate information from theexample satellite 100 of FIG. 1 to a target (e.g., the ground station104), receive signals (e.g., optical signals) from the target, generateimages of the target, etc. In the illustrated example, the payloadassembly 200 includes a base 202, a first actuator 204, a secondactuator 300 (FIG. 3), and a payload 206 such as, for example, anantenna, a sensor (e.g., an infrared sensor), a camera, an opticaldevice, etc. In the illustrated example, the base 202 is a triangularplate. Thus, the base 202 includes a first corner 208, a second corner210 and a third corner 212. The base 202 defines a substantially planarsurface 214 onto which the payload 206 is coupled. In the illustratedexample, the planar surface 214 is to substantially face the groundstation 104 during operation of the example satellite 100. In otherexamples, the base 202 is other shapes (e.g., rectangular, circular, ashape that defines a rounded or uneven (e.g., stepped) surface facingthe ground station 104, etc.). In the illustrated example, the payload206 is disposed on the base 202 substantially normal to the planarsurface 214 (i.e., the payload 206 is substantially perpendicular to thebase 202).

In the illustrated example, the base 202 is rotatably coupled to thesatellite 100 via a pivot joint 216 (e.g., a ball joint) disposed at oradjacent the first corner 208. The example joint 216 enables the base202 to rotate at point A about a first axis (X-axis) 218 and a secondaxis (Y-axis) 220. The example first axis 218 and the example secondaxis 220 intersect at point A. In the illustrated example, a third axis(Z-axis) 222 intersects the first axis 218 and the second axis 220 atpoint A. Thus, point A corresponds to coordinates (0, 0, 0). Theabove-noted axes 218, 220 and 222 are merely examples and, thus, otheraxes may be employed in other examples. In the illustrated example, anelevation angle, ε, of the base 202 is an amount of rotation of the base202 from coordinates (0, 0, 0) about the first axis 218. An azimuthangle, α, of the example base 202 is an amount of rotation of the base202 from coordinates (0, 0, 0) about the second axis 220. Thus, aposition or orientation of the example base 202 may be defined by theazimuth angle and the elevation angle of the base 202.

FIG. 3 is a rear view of the example payload assembly 200 of FIG. 2. Inthe illustrated example, the base 202 is operatively coupled to thesatellite 100 via the first actuator 204 and a second actuator 300,respectively. The example first actuator 204 is coupled to the base 202at point B, which is adjacent the second corner 210 of the base 202. Theexample second actuator 300 is coupled to the base 202 at point C, whichis adjacent the third corner 212 of the base 202. In the illustratedexample, the first actuator 204 and the second actuator 300 arejackscrews. However, the first actuator 204 and the second actuator 300may be implemented using any type of linear actuator. In the illustratedexample, strokes of the first actuator 204 and the second actuator 300are substantially parallel to the third axis 222. For example, whenactuated, a first arm 302 of the first actuator 204 and a second arm 304of the second actuator 300 move substantially parallel to the third axis222, thereby moving the second corner 210 and/or the third corner 212,respectively, toward or away from the satellite 100). The example firstactuator 204 and/or the example second actuator 300 may be actuated torotate the base 202 approximately ten degrees about an axis of rotation(e.g., the first axis 218 and/or the second axis 220). Other examplesrotate other amounts (e.g., five degrees, forty five degrees, etc.)and/or about other axes.

In the illustrated example, the first actuator 204 and the secondactuator 300 are in communication with a controller 306. The examplecontroller 306 may reside in the satellite 100, within the groundstation 104, and/or in any other suitable location. In the illustratedexample, the controller 306 controls a first stroke position, h₁, of thefirst actuator 204 and a second stroke position, h₂, of the secondactuator 300. In other examples, the first actuator 204 and the secondactuator 300 are controlled via separate controllers. As described ingreater detail below, the controller 306 determines and compensates fora first stroke position error corresponding to the first actuator 204and a second stroke position error corresponding to the second actuator300 to point the payload 206 at the ground station 104.

FIG. 4 is a top view of the example payload assembly 200 of FIGS. 2-3.In the illustrated example, point B (i.e., where the example firstactuator 204 is coupled to the base 202) is a first distance L1 frompoint A. Point C (i.e., where the example second actuator 300 is coupledto the base 202) is a second distance L2 from point A. In theillustrated example, the first distance L1 and the second distance L2are substantially equal. In other examples, the first distance L1 andthe second distance L2 may be different. The example first actuator 204and the example second actuator 300 are spaced apart from each othersuch that the first actuator 204 and the second actuator 300 aresubstantially ninety degrees apart relative to the joint 216 (i.e., aseparation angle ϕ between the first actuator 204 and the secondactuator 300 is substantially ninety degrees). In other examples, thefirst actuator 204 and/or the second actuator 300 are in other positionsrelative to the joint 216 (e.g., the separation angle ϕ is greater thanor less than ninety degrees, etc.).

In the illustrated example, to communicate information from the examplesatellite 100 to the ground station 104, the example satellite 100transmits one or more signals to the ground station 104 via the payload206. To facilitate transmission of the signal, the example payload 206is pointed at or toward the ground station 104. In other examples, thepayload 206 is pointed at a target other than the ground station 104 to,for example, generate images of the target, receive signals from thetarget, take measurements (e.g., via a sensor), etc. Thus, while thefollowing examples are described in conjunction with the example groundstation 104, the base may be oriented to point the payload 206 at anyother suitable target in accordance with the teachings of thisdisclosure. In the illustrated example, the payload 206 is pointed atthe ground station 104 by aligning the payload 206 with a line of sight(LOS) vector extending from the ground station 104 to the satellite 100or from the satellite 100 to the ground station 104 (e.g., such that thesignals transmitted via the payload 206 substantially propagate alongthe LOS vector). To align the payload 206 with the LOS vector, thecontroller 306 actuates the first actuator 204 and/or the secondactuator 300 to adjust an orientation of the base 202 (i.e., the azimuthangle and the elevation angle of the base 202) and, thus, a pointingdirection of the payload 206.

In some examples, the controller 306 and/or the ground station 104determine the LOS vector from the satellite 100 to the ground station104 and/or from the ground station 104 to the satellite 100. Based onthe LOS vector, an estimated base orientation (i.e., an estimatedazimuth angle and an estimated elevation angle of the base 202) to pointthe payload 206 along the LOS vector (e.g., such that a signaltransmitted via the payload propagates substantially along the LOSvector to the ground station 104) may be determined. To move (i.e.,orient) the base 202 to the estimated base orientation, a firstestimated stroke position and a second estimated stroke position of thefirst actuator 204 and the second actuator 300, respectively, aredetermined, and the first actuator 204 and the second actuator 300 areactuated to the first estimated stroke position and the second estimatedstroke position, respectively.

In some examples, the LOS vector is determined based on a position ofthe ground station 104 and a position of the satellite 100. The positionof the ground station 104 may be determined based on a position vectorof the ground station 104 in an inertial frame such as, for example, anEarth-Centered Earth Fixed frame based on an Earth-Centered InertialFrame. The position of the satellite 100 may be determined in a frame ofthe satellite 100 via an orbit frame. In some examples, the orbit frameis determined based on an orbit position vector of the satellite 100 inthe Earth-Centered Frame. The LOS vector may then be determined based ona difference vector between the position of the ground station 104 andthe position of the satellite 100 in the frame of the satellite 100.Based on the LOS vector, the estimated base orientation to point thepayload 206 at the ground station 104 may be determined.

In the illustrated example, once the estimated base orientation isdetermined, the first estimated stroke position of the first actuator204 and the second estimated stroke position of the second actuator 300to orient the base 202 at the estimated base orientation are determined.The first estimated stroke position, h₁ of the first actuator 204 andthe second estimated stroke position, h₂, of the second actuator 300 area function of the estimated azimuth angle, α, and the estimatedelevation angle, ε, of the base 202 as shown in the following equations:

$\begin{matrix}{\quad\{ {\begin{matrix}{{\tan\;\alpha} = \frac{( {h_{1} + h_{2}} )}{2L\;{\cos( {\phi\text{/}2} )}}} \\{{\tan\; ɛ} = {- \frac{\sqrt{2}{\cos( {\phi\text{/}2} )}( {h_{1} - h_{2}} )}{\sqrt{( {1 - {\cos(\phi)}} )\begin{pmatrix}{{2L^{2}} + {2L^{2}\cos(\phi)} +} \\( {h_{1} + h_{2}} )^{2}\end{pmatrix}}}}}\end{matrix}.} } & {{Equation}\mspace{11mu} s\mspace{14mu} 1\mspace{14mu}{and}\mspace{14mu} 2}\end{matrix}$

In Equations 1 and 2, ϕ is the separation angle, and L is the firstdistance L1 (i.e., the distance from point A to B) or the seconddistance L2 (i.e., the distance from point A to point C). In theillustrated examples disclosed herein, L1=L2. In other examples, when L1is not equal to L2, calculations and/or equations described hereinaccount for other variables. As a result, for example, Equations 1 and 2become more complex when L1 is not equal to L2. However, thecalculations and/or equations disclosed herein may be configured orobtained when L1 is not equal to L2. Equations 1 and 2 may be used foreach of the first and second actuators to determine the estimatedazimuth angle, α, and the estimated elevation angle, ε, of the base 202.In the illustrated example, because the separation angle ϕ issubstantially ninety degrees, Equation 1 and Equation 2 simplify asshown in Equations 3 and 4 below:

$\begin{matrix}{\phi =  90^{{^\circ}}arrow\{ {\begin{matrix}{{\tan\;\alpha} = \frac{\sqrt{2}( {h_{1} + h_{2}} )}{2L}} \\{{\tan\; ɛ} = {- \frac{( {h_{1} - h_{2}} )}{\sqrt{{2L^{2}} + ( {h_{1} + h_{2}} )^{2}}}}}\end{matrix}.}  } & {{Equation}\mspace{11mu} s\mspace{14mu} 3\mspace{14mu}{and}\mspace{14mu} 4}\end{matrix}$

Based on Equations 3 and 4, the first estimated stroke position, h₁, andthe second estimated stroke position, h₂, may be determined using thefollowing equations:

$\begin{matrix}\{ {\begin{matrix}{( {h_{1} + h_{2}} ) = \frac{2L\;\tan\;\alpha}{\sqrt{2}}} \\{( {h_{1} - h_{2}} ) = {{- \tan}\; ɛ\sqrt{{2L^{2}} + ( \frac{2L\;\tan\;\alpha}{\sqrt{2}} )^{2}}}}\end{matrix};{and}}  & {{Equation}\mspace{11mu} s\mspace{14mu} 5\mspace{14mu}{and}\mspace{14mu} 6} \\\{ \begin{matrix}{h_{1} = {\frac{1}{2}\begin{matrix}{( {\frac{2L\;\tan\;\alpha}{\sqrt{2}} - {\tan\; ɛ\sqrt{{2L^{2}} + ( \frac{2L\;\tan\;\alpha}{\sqrt{2}} )^{2}}}} ) =} \\{\frac{L}{\sqrt{2}}( {{\tan\;\alpha} - \frac{\tan\; ɛ}{\cos\;\alpha}} )}\end{matrix}}} \\{h_{2} = {\frac{1}{2}\begin{matrix}{( {\frac{2L\;\tan\;\alpha}{\sqrt{2}} + {\tan\; ɛ\sqrt{{2L^{2}} + ( \frac{2L\;\tan\;\alpha}{\sqrt{2}} )^{2}}}} ) =} \\{\frac{L}{\sqrt{2}}( {{\tan\;\alpha} + \frac{\tan\; ɛ}{\cos\;\alpha}} )}\end{matrix}}}\end{matrix}  & {{Equation}\mspace{11mu} s\mspace{14mu} 7\mspace{14mu}{and}\mspace{14mu} 8}\end{matrix}$Equations 7 and 8, for example, provide the actuators' travelcomputation. In some examples in which the separation angle ϕ is notninety degrees, the first estimated stroke position and the secondestimated stroke position may be determined using Equations 9-12 below:

$\quad\{ {\begin{matrix}{( {h_{1} + h_{2}} ) = {2L\;{\cos( {\phi\text{/}2} )}{\tan(\alpha)}}} \\{( {h_{1} - h_{2}} ) = {{- \frac{{\tan(ɛ)}\sqrt{\begin{matrix}( {1 - {\cos(\phi)}} ) \\\begin{pmatrix}{{2L^{2}} + {2L^{2}\cos(\phi)} +} \\( {2L\;{\cos( {\phi\text{/}2} )}{\tan(\alpha)}} )^{2}\end{pmatrix}\end{matrix}}}{\sqrt{2}{\cos( {\phi\text{/}2} )}}} = {- \frac{L\;{\tan(ɛ)}{\sin(\phi)}}{{\cos( {\phi\text{/}2} )}{\cos(\alpha)}}}}}\end{matrix}\{ {\begin{matrix}{h_{1} = {{\frac{1}{2}{L\begin{pmatrix}{{2\;{\cos( {\phi\text{/}2} )}\tan\;(\alpha)} -} \\\frac{{\tan(ɛ)}{\sin(\phi)}}{\;{{\cos( {\phi\text{/}2} )}{\cos(\alpha)}}}\end{pmatrix}}} = {L\begin{pmatrix}{\;{{{\cos( {\phi\text{/}2} )}\tan\;(\alpha)} -}} \\\frac{{\tan(ɛ)}{\sin( {\phi\text{/}2} )}}{\;{\cos(\alpha)}}\end{pmatrix}}}} \\{h_{2} = {{\frac{1}{2}{L\begin{pmatrix}{{2\;{\cos( {\phi\text{/}2} )}\tan\;(\alpha)} +} \\\frac{{\tan(ɛ)}{\sin(\phi)}}{\;{{\cos( {\phi\text{/}2} )}{\cos(\alpha)}}}\end{pmatrix}}} = {L\begin{pmatrix}{\;{{{\cos( {\phi\text{/}2} )}\tan\;(\alpha)} +}} \\\frac{{\tan(ɛ)}{\sin( {\phi\text{/}2} )}}{\;{\cos(\alpha)}}\end{pmatrix}}}}\end{matrix}.} } $

Using the first estimated stroke position and the second estimatedstroke position, the controller 306 communicates a command to the firstactuator 204 and the second actuator 300 to actuate to the firstestimated stroke position and the second estimated stroke position,respectively. In some examples, the payload 206 may not point at theground station 104 when the first actuator 204 and the second actuator300 actuate to the first estimate stroke position and the secondestimated stroke position, respectively. Instead, the base 202 may beoriented at a resultant base orientation different than the baseorientation at which the payload 206 points at the ground station 104. Adifference between the resultant base orientation and the baseorientation at which the payload 206 points to the ground station 104 isa base orientation or pointing error. If not compensated for, the baseorientation error may affect communication between the satellite 100 andthe ground station 104.

In the illustrated example, the base orientation error,

$\begin{bmatrix}{d\;\alpha} \\{d\; ɛ}\end{bmatrix},$is a function of a first stroke position error, dh₁, corresponding tothe first actuator 204 and a second stroke position error, dh₂,corresponding to the second actuator 300 as shown in the followingequation:

$\begin{matrix}{{{\begin{bmatrix}{d\;\alpha} \\{d\; ɛ}\end{bmatrix} = {{M\begin{bmatrix}{dh}_{1} \\{dh}_{2}\end{bmatrix}} = {\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix}\begin{bmatrix}{dh}_{1} \\{dh}_{2}\end{bmatrix}}}};{where}}{m_{11} = {m_{12} = \frac{\sqrt{2}L}{( {h_{1} + h_{2}} )^{2} + {2L^{2}}}}}{m_{21} = {{- m_{22}} = {- {\frac{{L^{3}( {L^{2} + h_{2}^{2} + {h_{1}h_{2}}} )}( {{2L^{2}} + ( {h_{1} + h_{2}} )^{2}} )}{( {L^{2} + h_{1}^{2} + h_{2}^{2}} )( {{L^{2}( {h_{1} + h_{2}} )}^{2} + {2L^{4}}} )^{3/2}}.}}}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$In some examples, the base orientation error is determinedexperimentally by, for example, communicating signals from the satellite100 to the ground station 104 and/or by communicating signals from theground station 104 to the satellite 100.

FIGS. 5-6 are plots 500 and 600 illustrating power levels of a payloadsignal detected at the ground station 104. FIG. 5 is a two-dimensionalplot 500 of the power levels detected at the ground station 104, andFIG. 6 is a three-dimensional plot 600 of the power levels detected atthe ground station 104. In the illustrated example, the base orientationerror is determined based on the power levels of the payload signaldetected at the ground station 104 while the payload 206 is moved toscan (e.g., raster scan) a given area. When the example payload 206 ispointed at the ground station 104, the ground station 104 detects amaximum power level. As the payload 206 points away from the groundstation 104, the ground station 104 detects a lesser power level. In theillustrated example, a first axis of each of the plots 500 and 600corresponds to the azimuth angle of the base 202, and a second axis ofeach of the plots 500 and 600 corresponds to the elevation of the base202. In the example plot 500 of FIG. 5, the power levels are illustratedby rings 502 having sizes corresponding to an amount of power detectedat the ground station 104. In the example plot of FIG. 5, a smallestring 504 corresponds to a maximum power level detected at the groundstation 104, and a largest ring 506 corresponds to a lowest power leveldetected at the ground station 104. In the example plot 600 of FIG. 6, athird axis corresponds to the power levels detected at the groundstation 104.

In the example plots 500 and 600 of FIGS. 5 and 6, coordinates (0, 0)and (0, 0, 0), respectively, correspond to the resultant baseorientation (i.e., an orientation of the base 202 when the firstactuator 204 and the second actuator 300 are actuated based on the firstestimated stroke position and the second estimated stroke position,respectively). When the base 202 is oriented at the resultant baseorientation, the ground station 104 detects the power levels of thesignal communicated by the payload 206. Then, the payload 206 is scanned(e.g., raster scanned) or moved to point to some or all positionsillustrated in the plots 500 and 600 of FIGS. 5-6 while the power levelsare detected (e.g., continuously, at predetermined intervals of time,etc.) at the ground station 104. In the illustrated example, the groundstation 104 detects the maximum power level when the base 202 ispositioned at coordinates (0.02, 0.04). Thus, the payload 206substantially points at the ground station 104 when the base 202 ispositioned at coordinates (0.02, 0.04). The base orientation error is adistance between the orientation of the base 202 at which the groundstation 104 measures the maximum amount of power (e.g., coordinates(0.02, 0.04)) and the resultant base orientation (e.g., (0, 0)) as shownin the following equation:

$\begin{matrix}{\begin{bmatrix}{d\;\alpha} \\{d\; ɛ}\end{bmatrix}_{measure} = {{\begin{bmatrix}{\;\alpha} \\{\; ɛ}\end{bmatrix}_{measure} - \begin{bmatrix}{\;\alpha} \\{\; ɛ}\end{bmatrix}_{cmd}}:}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

where the first term on the right labeled with subscript “measure”represents measured values of the base angles, the second term on theright labeled with subscript “cmd” represents the commanded values(e.g., estimated values) of the base angle, and the term on the left isthe difference between the commanded based angles and the measured baseangles, or the base orientation error.

In some examples, signals (e.g., radio frequency (RF) signals, etc.) arecommunicated from the ground station 104 (e.g., via ground-based beaconsystems) to the satellite 100 to determine the base orientation error.When the satellite 100 receives the signals, the satellite 100 decodesand/or demodulates the signals to determine the base orientation error.Other examples employ other techniques to determine the base orientationerror.

Based on the base orientation error and using, for example, thefollowing equation, a first stroke position error of the first actuator204 and a second stroke position error of the second actuator 300 may bedetermined:

$\begin{matrix}{{\begin{bmatrix}{dh}_{1} \\{dh}_{2}\end{bmatrix}_{est} = {\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix}^{- 1}\begin{bmatrix}{d\;\alpha} \\{d\; ɛ}\end{bmatrix}}_{measure}};} & {{Equation}\mspace{14mu} 15}\end{matrix}$

where the term on the left represents estimated first and second strokeposition errors and where

$m_{11} = {m_{12} = \frac{\sqrt{2}L}{( {h_{1} + h_{2}} )^{2} + {2L^{2}}}}$$m_{21} = {{- m_{22}} = {- \frac{{L^{3}( {L^{2} + h_{2}^{2} + {h_{1}h_{2}}} )}( {{2L^{2}} + ( {h_{1} + h_{2}} )^{2}} )}{( {L^{2} + h_{1}^{2} + h_{2}^{2}} )( {{L^{2}( {h_{1} + h_{2}} )}^{2} + {2L^{4}}} )^{3/2}}}}$when assuming L1 equals L2.

In the illustrated example, the first and second stroke position errors,dh₁ and dh₂, respectively, are inherent in the payload assembly 200(i.e., the stroke position errors are constant or invariable relative tothe orientation of the base 202 and/or an amount of rotation of the base202 via the first actuator 204 and/or the second actuator 300). Thus,the controller 306 may compensate for the first and second strokeposition errors to point the payload 206 at the target irrespective of arange of movement of the base 202. In the illustrated example, thecontroller 306 compensates for the first and second stroke positionerrors by commanding the first actuator 204 and the second actuator 300to actuate to the first and second corrected stroke positions. Todetermine the first and second corrected stroke positions, the first andsecond stroke position errors are subtracted from the first and secondestimated stroke positions, respectively, as shown in the followingequation:

$\begin{matrix}{\begin{bmatrix}h_{1} \\h_{2}\end{bmatrix} = {\underset{\begin{matrix}\begin{matrix}{{Nominally}\mspace{14mu}{calculated}\mspace{14mu}{actuator}} \\{{travel}\mspace{14mu}{based}\mspace{14mu}{on}\mspace{14mu}{the}\mspace{14mu}{desired}\mspace{14mu}{or}}\end{matrix} \\{{commanded}\mspace{14mu}{base}\mspace{14mu}{angles}}\end{matrix}}{\underset{︸}{\begin{bmatrix}h_{1} \\h_{2}\end{bmatrix}_{nonlinear}}} - \begin{bmatrix}{dh}_{1} \\{dh}_{2}\end{bmatrix}_{est}}} & {{Equation}\mspace{14mu} 16}\end{matrix}$

where the first term on the right labeled with subscript “nonlinear” isthe nominally calculated actuator travel based on the desired (orcommanded or estimated) base angles and the second term on the rightwith subscript “est” estimate the actuator errors. When the firstactuator 204 is actuated to the first corrected stroke position and thesecond actuator 300 is actuated to the second corrected stroke position,the base 202 is oriented such that the payload 206 points at the groundstation 104.

The pointing direction of the payload 206 may be subsequently adjustedto keep the payload 206 pointed at the ground station 104 (e.g., if thesatellite 100 moves relative to the ground station 104), point thepayload 206 at another target, etc. In such examples, the controller 306determines an updated estimated base orientation to point the payload206 at the ground station 104 (or other target) based on an updated LOSvector from the satellite 100 to the ground station 104 (or the othertarget) and/or from the ground station 104 to the satellite 100. Thecontroller 306 then determines a first updated estimated stroke positionand a second updated estimated stroke position based on the updatedestimated base orientation. To compensate for the first and secondstroke position errors, the controller 306 determines first and secondupdated corrected stroke positions. To determine the first and secondupdated corrected stroke positions, the controller 306 subtracts thefirst and second stroke position errors from the first and secondupdated estimated stroke positions, respectively. The controller 306communicates a command to the first actuator 204 and the second actuator300 to move to the first and second updated corrected stroke positions,and the first actuator 204 and the second actuator 300 actuate to pointthe payload 206 at the ground station 104 (or the other target).

In some examples, alignment errors such as, for example, actuatoralignment errors, actuator assembly parameter errors influenced by, forexample, thermal expansion, machining tolerances, etc. influence thefirst stroke position error and/or the second stroke position error. Insome such examples, the example controller 306 may compensate for thealignment errors. The example alignment errors determined below arerelated to the separation angle ϕ and the first distance L1 from point Ato point B and/or the second distance L2 from point A to point C.However, additionally or alternatively, other alignment errors may bedetermined in other examples. The example alignment errors may bedetermined using, for example, the following equation:

$\begin{matrix}{{{\begin{bmatrix}{d\;\alpha} \\{d\; ɛ}\end{bmatrix} = {{\lbrack {M\mspace{14mu} G} \rbrack\begin{bmatrix}{dh}_{1} \\{dh}_{1} \\{d\;\phi} \\{d\; L}\end{bmatrix}} = {{\underset{\underset{C}{︸}}{\begin{bmatrix}m_{11} & m_{12} & g_{11} & g_{12} \\m_{21} & m_{22} & g_{21} & g_{22}\end{bmatrix}}\begin{bmatrix}{dh}_{1} \\{dh}_{1} \\{d\;\phi} \\{d\; L}\end{bmatrix}} = C}}},{{{where}\mspace{14mu} m_{11}} = {m_{12} = \frac{\sqrt{2}L}{( {h_{1} + h_{2}} )^{2} + {2L^{2}}}}}}{m_{21} = {{- m_{22}} = {- \frac{{L^{3}( {L^{2} + h_{2}^{2} + {h_{1}h_{2}}} )}( {{2L^{2}} + ( {h_{1} + h_{2}} )^{2}} )}{( {L^{2} + h_{1}^{2} + h_{2}^{2}} )( {{L^{2}( {h_{1} + h_{2}} )}^{2} + {2L^{4}}} )^{3/2}}}}}{g_{11} = \frac{{\sin( \frac{\alpha}{2} )} \times ( {h_{1} + h_{2}} )}{4L \times {\cos^{2}( \frac{\alpha}{2} )} \times ( {\frac{( {h_{1} + h_{2}} )^{2}}{4L^{2} \times {\cos^{2}( \frac{\alpha}{2} )}} + 1} )}}{g_{12} = {- \frac{( {h_{1} + h_{2}} )}{2L^{2} \times {\cos( \frac{\alpha}{2} )} \times ( {\frac{( {h_{1} + h_{2}} )^{2}}{4L^{2} \times {\cos^{2}( \frac{\alpha}{2} )}} + 1} )}}}{g_{21} = {- \frac{\begin{matrix}{\frac{\sqrt{2}L\;{\cos( \frac{\alpha}{2} )}( {h_{1} - h_{2}} )}{{2\begin{bmatrix}{L^{4} - {L^{4}{\cos( {2\;\alpha} )}} -} \\{2{L^{2}( {h_{1} + h_{2}} )}^{2}( {\frac{\cos(\alpha)}{2} - \frac{1}{2}} )}\end{bmatrix}}^{1/2}} +} \\\frac{\sqrt{2}L\;{\cos( \frac{\alpha}{2} )}{( {h_{1} - h_{2}} )\begin{bmatrix}{{2L^{2}{\sin( {2\alpha} )}} +} \\{L^{2}{\sin(\alpha)}( {h_{1} + h_{2}} )^{2}}\end{bmatrix}}}{{2\begin{bmatrix}{L^{4} - {L^{4}{\cos( {2\;\alpha} )}} -} \\{2{L^{2}( {h_{1} + h_{2}} )}^{2}( {\frac{\cos(\alpha)}{2} - \frac{1}{2}} )}\end{bmatrix}}^{3/2}}\end{matrix}}{\frac{2L^{2}{\cos^{2}( \frac{\alpha}{2} )}( {h_{1} - h_{2}} )^{2}}{\begin{matrix}{{L^{4}{\cos( {2\;\alpha} )}} - L^{4} +} \\{2{L^{2}( {h_{1} + h_{2}} )}^{2}( {\frac{\cos(\alpha)}{2} - \frac{1}{2}} )}\end{matrix}} - 1}}}{g_{22} = {- \frac{\begin{matrix}{\frac{\sqrt{2}L\;{\cos( \frac{\alpha}{2} )}( {h_{1} - h_{2}} )}{\begin{bmatrix}{L^{4} - {L^{4}{\cos( {2\;\alpha} )}} -} \\{2{L^{2}( {h_{1} + h_{2}} )}^{2}( {\frac{\cos(\alpha)}{2} - \frac{1}{2}} )}\end{bmatrix}^{1/2}} +} \\\frac{\sqrt{2}L\;{\cos( \frac{\alpha}{2} )}{( {h_{1} - h_{2}} )\begin{bmatrix}{{4L^{3}{\cos( {2\alpha} )}} -} \\{{4L^{3}} + {4{L( {h_{1} + h_{2}} )}^{2}}} \\( {\frac{\cos(\alpha)}{2} - \frac{1}{2}} )\end{bmatrix}}}{{2\begin{bmatrix}{L^{4} - {L^{4}{\cos( {2\;\alpha} )}} -} \\{2{L^{2}( {h_{1} + h_{2}} )}^{2}( {\frac{\cos(\alpha)}{2} - \frac{1}{2}} )}\end{bmatrix}}^{3/2}}\end{matrix}}{\frac{2L^{2}{\cos^{2}( \frac{\alpha}{2} )}( {h_{1} - h_{2}} )^{2}}{\begin{matrix}{{L^{4}{\cos( {2\;\alpha} )}} - L^{4} +} \\{2{L^{2}( {h_{1} + h_{2}} )}^{2}( {\frac{\cos(\alpha)}{2} - \frac{1}{2}} )}\end{matrix}} - 1}}}} & {{Equation}\mspace{14mu} 17}\end{matrix}$

In Equation 17, L is the first distance L1 or the second distance L2when L1=L2. In some examples, when L1 is not equal to L2, equation 17becomes more complex and is not further described herein for simplicity.Thus, in some examples, a variation of Equation 17 may be used when L1is not equal to L2. Using, for example, Equation 18 below, a sequentialestimator may be used to estimate the alignment errors in Equation 17:

$\begin{matrix}{\begin{bmatrix}{dh}_{1} \\{dh}_{1} \\{d\;\phi} \\{d\; L}\end{bmatrix}_{{est},k} = {{\overset{\_}{x}}_{k} = {{\overset{\_}{x}}_{k - 1} + {K_{k}( {\begin{bmatrix}{d\;\alpha} \\{d\; ɛ}\end{bmatrix}_{measure} - {C\;{\overset{\_}{x}}_{k - 1}}} )}}}} & {{Equation}\mspace{14mu} 18}\end{matrix}$where the subscript “est” represent the estimated or commanded valuesand the subscript “measure” represent the measured values. In otherexamples, other estimators are used. In Equation 18, K is an update gainmatrix at a k-th step. The gain matrix may be designed using anysuitable method(s) such as minimum-variance, Kalman filter, fixed gainobservers, etc. Thus, in the illustrated example, an amount of thermaldistortion along the first distance (or any other suitable portion ofthe base 202) may be determined. Based on the alignment errorsdetermined using Equations 17 and 18, corrected actuator assemblyparameter values (e.g., the separation angle ϕ, the first distance L1,etc.) may be determined as follows:

$\begin{matrix}\{ \begin{matrix}{\phi_{est} = {\phi + {d\;\phi_{est}}}} \\{L_{est} = {L + {d\; L_{est}}}}\end{matrix}  & {{Equation}\mspace{14mu} 19}\end{matrix}$where values with subscript “est” represent estimated values. Becausethe alignment errors affect a determination of the estimated strokepositions, corrected estimated stroke positions may be determined basedon the corrected actuator assembly parameters values as show in Equation20 below:

$\begin{matrix}{\begin{bmatrix}h_{1} \\h_{2}\end{bmatrix}_{{cmd}\;\_\;{nom}} = \underset{\begin{matrix}\begin{matrix}{{Calculated}\mspace{14mu}{actuator}\mspace{14mu}{steps}\mspace{14mu}{from}\mspace{14mu}{nonlinear}\mspace{14mu}{relation}} \\{{of}\mspace{14mu}{required}\mspace{14mu}{base}\mspace{14mu}{angles}\mspace{14mu}{and}\mspace{14mu}{the}\mspace{14mu}{estimated}}\end{matrix} \\{{actuator}\mspace{14mu}{geometry}\mspace{14mu}{parameters}}\end{matrix}}{\underset{︸}{\begin{bmatrix}{h_{1}( {\alpha,ɛ,\phi_{est},L_{est}} )} \\{h_{2}( {\alpha,ɛ,\phi_{est},L_{est}} )}\end{bmatrix}_{nonlinear}}}} & {{Equation}\mspace{14mu} 20}\end{matrix}$Based on the corrected estimated stroke positions and the strokeposition errors, the corrected stroke positions may be determined usingthe following equation:

$\begin{matrix}{\begin{bmatrix}h_{1} \\h_{2}\end{bmatrix}_{cmd} = {\begin{bmatrix}{h_{1}( {\alpha,ɛ,\phi_{est},L_{est}} )} \\{h_{2}( {\alpha,ɛ,\phi_{est},L_{est}} )}\end{bmatrix}_{nonlinear} - \begin{bmatrix}{dh}_{1} \\{dh}_{2}\end{bmatrix}_{est}}} & {{Equation}\mspace{14mu} 21}\end{matrix}$Thus, once the corrected stroke positions are determined, the controller306 may communicate with the first actuator 204 and the second actuator300 to actuate to the corrected stroke positions to point the payload206 at the ground station 104. In such examples, the corrected strokepositions compensate for the stroke position errors and the alignmenterrors.

FIG. 7 is a block diagram of the example controller 306 of FIG. 3. Inthe illustrated example, the controller 306 includes an instructionprocessor 700, a LOS information determiner 702, a base orientationdeterminer 704, a memory 706, an estimated stroke position determiner708, a base orientation error determiner 710, a stroke position errordeterminer 712, an alignment error determiner 714, a corrected strokeposition determiner 716, and an actuator controller 718.

The example instruction processor 700 of FIG. 7 receives instructionsfrom the ground station 104 and/or a flight computer 720 disposed onand/or in communication with the satellite 100. In some examples, theinstructions include a command to determine a, LOS vector to a target,adjust a pointing direction of the payload 206, actuate the firstactuator 204 and/or the second actuator 300, transmit a signal to theground station 104 via the payload 206, decode and/or demodulate signalsreceived via the satellite 100, etc. These instructions may be executedvia the example controller 306, stored in the memory 706, communicatedto one or more components of the satellite 100 and/or the ground station104, etc.

The example LOS determiner of FIG. 7 determines the LOS vector betweenthe satellite 100 and a target (e.g., the ground station 104). In someexamples, the LOS vector is based on the position of the satellite 100and the position of the target in one or more frames of reference. TheLOS information determiner 702 may access and/or utilize positioninformation related to the satellite 100 and/or the target from thememory 706 and/or the flight computer 720. In some examples, in responseto the flight computer 720 indicating relative movement between thesatellite 100 and the ground station 104, instructions to point thepayload 206 at a different target, etc., the LOS information determiner702 determines an updated LOS vector from the satellite 100 to thetarget and/or from the target to the satellite 100. In some examples,some or all of the LOS information is determined and/or provided via theflight computer and/or the target.

The example base orientation determiner 704 of FIG. 7 determines theestimated base orientation to point the payload 206 at the target (e.g.,orient the payload 206 to enable signals transmitted and/or received viathe payload 206 to propagate substantially along the LOS vector to thetarget). In the illustrated example, the base orientation determiner 704determines an estimated azimuth angle and an estimated elevation angleof the base 202 to point the payload 206 at the target.

The estimated stroke position determiner 708 determines a firstestimated stroke position and a second estimated stroke position of thefirst actuator 204 and the second actuator 300, respectively, to orientthe base 202 at the estimated base orientation. In some examples, theestimated stroke position determiner 708 determines the estimated strokepositions based on the estimated base orientation by, for example, usingEquations 1-8 and/or Equations 9-12 above. If the example alignmenterror determiner 714 determines that one or alignment errors arepresent, the estimated stroke position determiner 708 determinescorrected estimated stroke positions to compensate for the alignmenterrors.

The example base orientation error determiner 710 of FIG. 7 determinesthe base orientation error. In some examples, the base orientation errordeterminer 710 determines the base orientation error experimentally. Forexample, the base orientation error determiner 710 may instruct atransmitter 722 (e.g., an antenna transmitter) to transmit a signal whenthe first actuator 204 is in the first estimated stroke position and thesecond actuator 300 is in the second estimated stroke position. At theground station 104, a power level of the signal is detected, and thebase orientation error determiner 710 may instruct the actuatorcontroller 718 to move (e.g., span) the payload 206. At the groundstation 104, power levels of the signal are detected while the payload206 is moved. Based on the power levels detected at the ground station104, the base orientation error is determined by the base orientationerror determiner 710. In some examples, the base orientation error isdetermined via the ground station 104 and communicated to satellite 100.In other examples, signals (e.g., radio frequency (RF) signals, etc.)are communicated from the ground station 104 (e.g., via ground-basedbeacon systems) to the satellite 100, and the base orientation errordeterminer 710 determines the base orientation error by decoding and/ordemodulating the signals.

The example stroke position error determiner 712 of FIG. 7 determines afirst stroke position error of the first actuator 204 and a secondstroke position error of the second actuator 300. In the illustratedexample, the stroke position error determiner 712 determines the firstand second stroke position errors based on the base orientation error.In the illustrated example, the stroke position errors are inherent inthe payload assembly 200 (i.e., the stroke position errors are constantor invariable relative to the orientation of the base 202 and/or anamount of rotation of the base 202 via the first actuator 204 and/or thesecond actuator 300). Thus, the controller 306 may compensate for thefirst and second stroke position errors irrespective of an amount ofmovement of the base 202 to point the payload 206 at the target.

The example alignment error determiner 714 of FIG. 7 determinesalignment errors of the example actuator assembly such as, for example,an alignment error of the first actuator 204 and/or the second actuator300, actuator assembly parameter value errors influenced by thermaldistortion (e.g., thermal expansion, thermal contraction, bendinginfluenced by a temperature gradient, etc.), machining tolerances, etc,and/or other alignment errors, etc. In some examples, the alignmenterror determiner 714 determines corrected actuator assembly parametervalues such as, for example, a corrected separation angle, a correcteddistance between point A and point C (i.e., a length between a firstpoint about which the base 202 rotates via the joint 216 and a secondpoint where the second actuator 300 is coupled to the base 202). Thecorrected actuator assembly parameter values may be used by theestimated stroke position determiner 708 to determine correctedestimated stroke positions.

The example corrected stroke position determiner 716 of FIG. 7determines a first corrected stroke position of the first actuator 204and a second corrected stroke position of the second actuator 300 basedon the first and second stroke position errors, respectively. In someexamples, the first corrected stroke position and the second correctedstroke position compensate for the first and second stroke positionerrors, respectively, and the alignment errors. When the first actuator204 and the second actuator 300 are actuated to the first correctedstroke position and the second corrected stroke position, respectively,the payload 206 points at the target.

The actuator controller 718 controls the first actuator 204 and/or thesecond actuator 300. In the illustrated example, the actuator controller718 instructs the first actuator 204 to actuate to a given strokeposition such as, for example, the first estimated stroke position, thefirst corrected stroke position, and/or any other stroke position. Theexample actuator controller 718 may instruct the second actuator 300 toactuate to a given stroke position such as, for example, the secondestimated stroke position, the second corrected stroke position, and/orany other stroke position. In some examples, the actuator controller 718instructs the first actuator 204 and/or the second actuator 300 to movethe payload 206 to scan (e.g., raster scan) an area.

The example memory 706 (e.g., volatile memory, non-volatile memory,etc.) stores information such as, for example, a position of the groundstation 104, a position of the satellite 100, actuator assemblyparameter values, corrected actuator assembly parameter values,estimated stroke positions, stroke position errors, payload alignmenterrors, and/or any other information. The information stored in theexample memory 706 may be accessed by one or more components of theexample controller 306, the example satellite 100, the example groundstation 104, etc.

While an example manner of implementing the controller 306 of FIG. 3 hasbeen illustrated in FIG. 7, one or more of the elements, processesand/or devices illustrated in FIG. 7 may be combined, divided,re-arranged, omitted, eliminated and/or implemented in any other way.Further, the instruction processor 700, the LOS information determiner702, the base orientation determiner 704, the memory 706, the estimatedstroke position determiner 708, the base orientation error determiner710, the stroke position error determiner 712, the alignment errordeterminer 714, the corrected stroke position determiner 716, theactuator controller 718, the ground station 104, the flight computer720, the transmitter 722, the first actuator 204, the second actuator300 and/or, more generally, the example controller 306 of FIG. 7 may beimplemented by hardware, software, firmware and/or any combination ofhardware, software and/or firmware. Thus, for example, any of theinstruction processor 700, the LOS information determiner 702, the baseorientation determiner 704, the memory 706, the estimated strokeposition determiner 708, the base orientation error determiner 710, thestroke position error determiner 712, the alignment error determiner714, the corrected stroke position determiner 716, the actuatorcontroller 718, the ground station 104, the flight computer 720, thetransmitter 722, the first actuator 204, the second actuator 300 and/or,more generally, the example controller 306 of FIG. 7 could beimplemented by one or more circuit(s), programmable processor(s),application specific integrated circuit(s) (ASIC(s)), programmable logicdevice(s) (PLD(s)) and/or field programmable logic device(s) (FPLD(s)),etc. When any of the apparatus or system claims of this patent are readto cover a purely software and/or firmware implementation, at least oneof the instruction processor 700, the LOS information determiner 702,the base orientation determiner 704, the memory 706, the estimatedstroke position determiner 708, the base orientation error determiner710, the stroke position error determiner 712, the alignment errordeterminer 714, the corrected stroke position determiner 716, theactuator controller 718, the ground station 104, the flight computer720, the transmitter 722, the first actuator 204, the second actuator300 and/or, more generally, the example controller 306 of FIG. 7 arehereby expressly defined to include a tangible computer readable mediumsuch as a memory, DVD, CD, Blu-ray, etc. storing the software and/orfirmware. Further still, the example controller 306 of FIG. 7 mayinclude one or more elements, processes and/or devices in addition to,or instead of, those illustrated in FIG. 7, and/or may include more thanone of any or all of the illustrated elements, processes and devices.

FIGS. 8-9 depict example flow diagrams representative of methods orprocesses that may be implemented using, for example, computer readableinstructions. The example processes of FIGS. 8-9 may be performed usinga processor, a controller (e.g., the example controller 306 of FIG. 7)and/or any other suitable processing device. For example, the exampleprocesses of FIGS. 8-9 may be implemented using coded instructions(e.g., computer readable instructions) stored on a tangible computerreadable medium such as a flash memory, a read-only memory (ROM), and/ora random-access memory (RAM). As used herein, the term tangible computerreadable medium is expressly defined to include any type of computerreadable storage and to exclude propagating signals. Additionally oralternatively, the example process of FIGS. 8-9 may be implemented usingcoded instructions (e.g., computer readable instructions) stored on anon-transitory computer readable medium such as a flash memory, aread-only memory (ROM), a random-access memory (RAM), a cache, or anyother storage media in which information is stored for any duration(e.g., for extended time periods, permanently, brief instances, fortemporarily buffering, and/or for caching of the information). As usedherein, the term non-transitory computer readable medium is expresslydefined to include any type of computer readable medium and to excludepropagating signals.

Alternatively, some or all of the example processes of FIGS. 8-9 may beimplemented using any combination(s) of application specific integratedcircuit(s) (ASIC(s)), programmable logic device(s) (PLD(s)), fieldprogrammable logic device(s) (FPLD(s)), discrete logic, hardware,firmware, etc. Also, one or more operations depicted in FIGS. 8-9 may beimplemented manually or as any combination(s) of any of the foregoingtechniques, for example, any combination of firmware, software, discretelogic and/or hardware.

Further, although the example processes of FIGS. 8-9 are described withreference to the flow diagrams of FIG. 8-9, respectively, other methodsof implementing the processes of FIGS. 8-9 may be employed. For example,the order of execution of the blocks may be changed, and/or some of theblocks described may be changed, eliminated, sub-divided, or combined.Additionally, one or more of the operations depicted in FIGS. 8-9 may beperformed sequentially and/or in parallel by, for example, separateprocessing threads, processors, devices, discrete logic, circuits, etc.

FIG. 8 is a flowchart representative of an example method 800 that canbe performed to point the payload 206 at a target such as, for example,the ground station of FIG. 1. The example method 800 of FIG. 8 begins bythe LOS information determiner 702 of the controller 306 determiningline of sight information (block 802). In some examples, determining theline of sight information includes determining a line of sight vectorfrom the target (e.g., the ground station 104) to the satellite 100and/or from the satellite 100 to the target. In some examples, theground station 104 and/or the flight computer 720 determines and/orprovides some or all of the LOS information. Based on the line of sightinformation, the base orientation determiner 704 determines an estimatedbase orientation to point the payload 206 at the target (e.g., theground station 104) (block 804). In some examples, the estimated baseorientation is defined by an estimated azimuth angle of the base 202 andan estimated elevation angle of the base 202 to point the payload 206 atthe target.

At block 806, the estimated stroke position determiner 708 determines afirst estimated stroke position of the first actuator 204 and a secondestimated stroke position of the second actuator 300 to orient the base202 at the estimated base orientation. At block 808, the actuatorcontroller 718 communicates a command to the first actuator 204 and thesecond actuator 300 to actuate to the first estimated stroke positionand the second estimated stroke position, respectively. In someexamples, as a result of thermal distortion, machining tolerances, etc.,the payload 206 does not point at the target when the first actuator 204and the second actuator 300 actuate to the first estimated strokeposition and the second estimated stroke position, respectively. In suchexamples, the base 202 moves to a resultant base orientation. At block810, the base orientation error determiner 710 determines a baseorientation error. The base orientation error may be a distance from theresultant base orientation to the position at which the payload 206points at the target. In some examples, the base orientation error isdetermined by detecting power levels of a payload signal (e.g.,transmitted via the transmitter 722) at the target. In other examples,the base orientation error is determined by decoding and/or demodulatingsignals received via the satellite 100 and/or via any other suitabletechnique.

At block 812, the alignment error determiner 714 determines an alignmenterror of the base 202. In some examples, the alignment error includes anerror in an alignment of the first actuator 204 and/or the secondactuator 300, an error in an actuator assembly parameter value such as,for example, the separation angle ϕ, the first distance L1, the seconddistance L2, etc. In some examples, the alignment error determiner 714determines one or more corrected actuator assembly parameter valuesbased on the alignment error. Based on the base orientation error andthe alignment error, the stroke position error determiner 712 determinesa first stroke position error and a second stroke position errorcorresponding to the first actuator 204 and the second actuator 300,respectively (block 814). At block 816, the corrected stroke positiondeterminer 716 determines a first corrected stroke position and a secondcorrected stroke position based on the first stroke position error andthe second stroke position error, respectively. In the illustratedexample, the first and second corrected stroke positions compensate forthe alignment error and the stroke position errors. At block 818, theactuator controller 718 communicates a command to the first actuator 204and the second actuator 300 to actuate to the first corrected strokeposition and the second corrected stroke position, respectively. Whenthe first actuator 204 and the second actuator 300 actuate to the firstcorrected stroke position and the second corrected stroke position,respectively, the first actuator 204 and the second actuator 300 orientthe base 202 such that the payload 206 points at the target.

FIG. 9 is a flowchart representative of an example method 900 that canbe performed to point the payload 206 at a target. The example method900 of FIG. 9 begins by determining if the payload pointing direction isto be adjusted (block 902). In some examples, the flight computer 720monitors a position of the satellite 100 relative to the target. If theflight computer 720 senses relative movement of the satellite 100, theflight computer 720 communicates instructions to the instructionprocessor 700 to adjust a pointing direction of the payload 206 to pointthe payload 206 at the target. In some examples, the controller 306 maybe instructed to adjust the pointing direction of the payload 206 topoint the payload 206 at a different target. If the payload pointingdirection is to be adjusted, the LOS information determiner 702determines line of sight information (block 904). In some examples, theflight computer 720 and/or the target provides and/or determines some orall of the LOS information. In some examples, the line of sightinformation includes a line of sight vector from the target to thesatellite 100 and/or from the satellite 100 to the target. Based on thelight of sight information, the base orientation determiner 704determines the estimated base orientation to point the payload 206 atthe target (block 906). At block 908, the estimated stroke positiondeterminer 708 determines a first estimated stroke position of the firstactuator 204 and a second estimated stroke position of the secondactuator 300 to orient the base 202 at the estimated base orientation.In some examples, the first estimated stroke position and/or the secondestimated stroke position may be determined based on previouslydetermined corrected actuator assembly parameter values such as, forexample, a corrected separation angle, etc. The previously determinedcorrected actuator assembly parameter values accessed via the memory706.

At block 910, the corrected stroke position determiner 716 determines afirst corrected stroke position and a second corrected stroke positionof the first actuator 204 and the second actuator 300, respectively. Todetermine the first corrected stroke position, the example correctedstroke position determiner 716 compensates for a previously determinedfirst stroke position error (e.g., by subtracting the previouslydetermined first stroke position error from the first estimated strokeposition). To determine the second corrected stroke position, theexample corrected stroke position determiner 716 compensates for apreviously determined second stroke position error (e.g., by subtractingthe previously determined second stroke position error from the secondestimated stroke position).

At block 912, the actuator controller 718 communicates a command to thefirst actuator 204 and the second actuator 300 to actuate to the firstcorrected stroke position and the second corrected stroke position,respectively. When the first actuator 204 and the second actuator 300are actuated to the first corrected stroke position and the secondcorrected stroke position, respectively, the base 202 is oriented suchthat the payload 206 points at the target.

FIG. 10 is a block diagram of an example computer 1000 capable ofexecuting the instructions of FIGS. 8-9 to implement the controller 306of FIG. 7. The computer 1000 can be any suitable type of computingdevice.

The computer 1000 of the instant example includes a processor 1012. Forexample, the processor 1012 can be implemented by one or moremicroprocessors or controllers from any desired family or manufacturer.

The processor 1012 includes a local memory 1013 (e.g., a cache) and isin communication with a main memory including a volatile memory 1014 anda non-volatile memory 1016 via a bus 1018. The volatile memory 1014 maybe implemented by Synchronous Dynamic Random Access Memory (SDRAM),Dynamic Random Access Memory (DRAM), RAMBUS Dynamic Random Access Memory(RDRAM) and/or any other type of random access memory device. Thenon-volatile memory 1016 may be implemented by flash memory and/or anyother desired type of memory device. Access to the main memory 1014,1016 is controlled by a memory controller.

The computer 1000 also includes an interface circuit 1020. The interfacecircuit 1020 may be implemented by any type of interface standard, suchas an Ethernet interface, a universal serial bus (USB), and/or a PCIexpress interface.

One or more input devices 1022 are connected to the interface circuit1020. The input device(s) 1022 permit a user to enter data and commandsinto the processor 1012.

One or more output devices 1024 are also connected to the interfacecircuit 1020. The output devices 1024 can be implemented, for example,by a transmitter (e.g., the transmitter 722). The interface circuit1020, thus, may include a graphics driver card.

The interface circuit 1020 also includes a communication device (e.g.,communication device 56) such as a modem or network interface card tofacilitate exchange of data with external computers via a network 1026(e.g., a bus, coaxial cable, RF signal transmitter, etc.).

The computer 1000 also includes one or more mass storage devices 1028for storing software and data. Examples of such mass storage devices1028 include hard drive disks, compact disk drives and digital versatiledisk (DVD) drives. The mass storage device 1028 may implement the localstorage device 62.

The coded instructions 1032 of FIG. 10 may be stored in the mass storagedevice 1028, in the volatile memory 1014, in the non-volatile memory1016, and/or on a removable storage medium such as a CD or DVD.

Although certain example methods, apparatus and articles of manufacturehave been described herein, the scope of coverage of this disclosure isnot limited thereto. On the contrary, this disclosure covers allmethods, apparatus and articles of manufacture fairly falling within thescope of the claims.

The Abstract at the end of this disclosure is provided to comply with 37C.F.R. § 1.72(b) to allow the reader to quickly ascertain the nature ofthe technical disclosure. It is submitted with the understanding that itwill not be used to interpret or limit the scope or meaning of theclaims.

What is claimed is:
 1. A non-transitory computer-readable medium havinginstructions stored thereon that, when executed, cause a machine to atleast: estimate a target pointing direction of a base supporting apayload to point the payload at a target, wherein the base is movablerelative to a pivot joint about an azimuth angle and an elevation anglevia a first linear actuator and a second linear actuator directlycoupled to the base; determine a first corrected stroke position of thefirst linear actuator based on a base orientation error, and a secondcorrected stroke position of the second linear actuator based on thebase orientation error; and command the first linear actuator to move tothe first corrected stroke position and the second linear actuator tomove to the second corrected stroke position to point the payload at thetarget without verifying a final position of the payload after the firstlinear actuator is actuated to the first corrected stroke position andthe second linear actuator is actuated to the second corrected strokeposition and without using a feedback to verify the base being at thetarget pointing direction when the first and second linear actuators arepositioned to the respective first and second corrected strokepositions.
 2. The non-transitory computer-readable medium as defined inclaim 1 comprising instructions that, when executed, cause the machineto determine a first estimated stroke position of the first linearactuator and a second estimated stroke position of the second linearactuator to point the base at the estimated target pointing direction.3. The non-transitory computer-readable medium as defined in claim 2comprising instructions that, when executed, cause the machine todetermine a first stroke position error of the first linear actuatorbased on the base orientation error and a second stroke position errorof the second linear actuator based on the base orientation error. 4.The non-transitory computer-readable medium as defined in claim 3comprising instructions that, when executed, cause the machine todetermine the first corrected stroke position of the first linearactuator based on a difference between the first stroke position errorand the first estimated stroke position.
 5. The non-transitorycomputer-readable medium as defined in claim 4 comprising instructionsthat, when executed, cause the machine to determine the second correctedstroke position of the second linear actuator based on a differencebetween the second stroke position error and the second estimated strokeposition.
 6. The non-transitory computer-readable medium as defined inclaim 2 comprising instructions that, when executed, cause the machine,when determining the base orientation error, to: determine a resultantbase orientation when the first actuator is positioned at the firstestimated stroke position and the second actuator is positioned at thesecond estimated stroke position; communicate a first signal between thepayload and the target when the base is at the resultant baseorientation; detect a power level of the first signal; communicate atest signal between the payload and the target when the base ispositioned a plurality of test orientations; detect a maximum powerlevel of the test signals; associate a test orientation of the base withthe detected maximum power level of the test signal; and determine thebase orientation error as a difference between the test orientation ofthe base associated with the detected maximum power level of the signaland the resultant base orientation.
 7. The non-transitorycomputer-readable medium as defined in claim 1 comprising instructionsthat, when executed, cause the machine to retrieve an experimentallydetermined value representative of the base orientation error.
 8. Anon-transitory computer-readable medium having instructions storedthereon that, when executed, cause a machine to at least: receive afirst command to move a payload relative to a first target; estimate abase orientation of a base coupled to the payload to point the payloadto the first target along a first line of sight vector; determine afirst estimated stroke position of a first linear actuator and a secondestimated stroke position of a second linear actuator to position thebase to the estimated base orientation; modify the first estimatedstroke position to a first corrected stroke position based on a baseorientation error; modify the second estimated stroke position to asecond corrected stroke position based on the base orientation error;and command the first linear actuator to actuate to the first correctedstroke position and command the second linear actuator to actuate to thesecond corrected stroke position without verifying a final position ofthe payload after the first linear actuator is actuated to the firstcorrected stroke position and the second linear actuator is actuated tothe second corrected stroke position.
 9. The non-transitorycomputer-readable medium as defined in claim 8 comprising instructionsthat, when executed, cause the machine to determine a difference betweenthe first estimated stroke position and a first stroke position error ofthe first linear actuator, wherein the first stroke position error isprovided by the base orientation error when determining the firstcorrected stroke position.
 10. The non-transitory computer-readablemedium as defined in claim 9 comprising instructions that, whenexecuted, cause the machine to determine a difference between the secondestimated stroke position and a second stroke position error of thesecond linear actuator, wherein the second stroke position error isprovided by the base orientation error when determining the secondcorrected stroke position.
 11. The non-transitory computer-readablemedium as defined in claim 8 comprising instructions that, whenexecuted, cause the machine, when determining the base orientationerror, to: receive a second command to orient the payload to a secondtarget prior to receiving the first command; determine a third estimatedstroke position of the first linear actuator and a fourth estimatedstroke position of the second linear actuator to point the payload tothe second target along a second line of sight vector; command the firstactuator to actuate to the third estimated stroke position and thesecond actuator to actuate to the fourth estimated stroke position;determine a resultant base orientation of the base when the first andsecond actuators are actuated to the third and fourth estimated strokepositions; detect a power level of a signal communicated by the payloadwhen the base is at the resultant base orientation; actuate the firstand second linear actuators to move the payload to scanning positionsdifferent than the resultant base orientation; measure additional powerlevels of signals communicated by the payload when the base ispositioned at each of the scanning positions; compare all the powerlevels of the signals associated with the scanning positions and theresultant base orientation; detect a maximum power level and anorientation of the base associated with the detected maximum powerlevel; and determine the base orientation error by measuring adifference between the resultant base orientation and the orientation ofthe base associated with the detected maximum power level.
 12. Thenon-transitory computer-readable medium as defined in claim 8 comprisinginstructions that, when executed, cause the machine to obtain an updatedline of sight vector between the payload and a known target.
 13. Thenon-transitory computer-readable medium as defined in claim 12comprising instructions that, when executed, cause the machine to:determine an updated estimated base orientation to point the payloadalong the updated line of sight vector; determine a first updatedestimated stroke position and a second updated estimated stroke positionbased on the updated line of sight vector; modify the first updatedestimated stroke position to a first updated corrected stroke positionbased on the base orientation error; modify the second updated estimatedstroke position to a second updated corrected stroke position based onthe base orientation error; and command the first actuator and thesecond actuator to move to the respective first and second updatedcorrected stroke positions.
 14. An apparatus to control a satellite, theapparatus comprising: a processor configured to: determine a targetpointing direction of a base to point a payload coupled to the base at atarget along a line of sight vector; determine a first estimated strokeposition of a first linear actuator and a second estimated strokeposition of a second linear actuator to point the base at the targetpointing direction; determine a first corrected stroke position of thefirst linear actuator based on a base orientation error, and determine asecond corrected stroke position of the second linear actuator based onthe base orientation error; and an actuator controller to command thefirst linear actuator to move to the first corrected stroke position andthe second linear actuator to move to the second corrected strokeposition to point the payload at the target along the line of sightvector without verifying the target pointing direction of the base whenthe first and second linear actuators are positioned to the respectivefirst and second corrected stroke positions and without using a feedbackto verify the base being at the target pointing direction when the firstand second linear actuators are positioned to the respective first andsecond corrected stroke positions.
 15. The apparatus of claim 14,wherein the processor is to determine a first stroke positioner error ofthe first linear actuator based on the base orientation error and asecond stroke position error of the second linear actuator based on thebase orientation error.
 16. The apparatus of claim 15, wherein theprocessor determines the first corrected stroke position based on adifference between the first stroke position error and the firstestimated stroke position, and determines the second corrected strokeposition based on a difference between the second stroke position errorand the second estimated stroke position.
 17. The apparatus of claim 14,further including an instruction processor to receive a command to pointthe payload at the target, wherein the base is movable relative to apivot about an azimuth angle and an elevation angle via only the firstlinear actuator and the second linear actuator, wherein the pivot, thefirst linear actuator and the second linear actuator are directlycoupled to the base.
 18. The apparatus of claim 14, wherein theprocessor is to determine the line of sight vector between the payloadand the target.
 19. The apparatus of claim 14, wherein the processor isto determine the base orientation error.
 20. The apparatus of claim 19,wherein the processor is to determine the base orientation errorexperimentally.
 21. The apparatus of claim 19, wherein the processor isto: determine a resultant base orientation when the first linearactuator is positioned at a third estimated stroke position and thesecond linear actuator is positioned at a fourth estimated strokeposition; communicate a first signal between the payload and the targetwhen the base is at the resultant base orientation; detect a power levelof the first signal; communicates a test signal between the payload andthe target when the base is positioned a plurality of test orientations;detect a maximum power level of the test signals; associate a testorientation of the base with the detected maximum power level of thetest signal; and determine the base orientation error as a differencebetween the test orientation of the base associated with the detectedmaximum power level of the signal and the resultant base orientation.